Small group number 237 of order 64

G is the group 64gp237

The Hall-Senior number of this group is 168.

G has 4 minimal generators, rank 3 and exponent 4. The centre has rank 2.

There are 2 conjugacy classes of maximal elementary abelian subgroups. Their ranks are: 3, 3.

This cohomology ring calculation is complete.

Ring structure | Completion information | Koszul information | Restriction information | Poincaré series

Ring structure

The cohomology ring has 7 generators:

There are 6 minimal relations:

A minimal Gröbner basis for the relations ideal consists of this minimal generating set, together with the following redundant relations:

Completion information

This cohomology ring was obtained from a calculation out to degree 12. The cohomology ring approximation is stable from degree 8 onwards, and Benson's tests detect stability from degree 8 onwards.

This cohomology ring has dimension 3 and depth 2. Here is a homogeneous system of parameters:

The first 2 terms h1, h2 form a regular sequence of maximum length.

The first 2 terms h1, h2 form a complete Duflot regular sequence. That is, their restrictions to the greatest central elementary abelian subgroup form a regular sequence of maximal length.

Data for Benson's test:

Koszul information

A basis for R/(h1, h2, h3) is as follows.

A basis for AnnR/(h1, h2)(h3) is as follows.

Restriction information